Periodic points of complex quadratic mappings

This article describes periodic points of some complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that involves the previous value raised to the powers one and two; and a complex map is one in which the variable and the parameters are complex numbers. A periodic point of a map is a value of the variable that occurs repeatedly after intervals of a fixed length. These periodic points play a role in the theories of Fatou and Julia sets. Let be the complex quadric mapping, where and are complex numbers. Notationally, is the -fold composition of with itself (not to be confused with the th derivative of )—that is, the value after the k-th iteration of the function Thus Periodic points of a complex quadratic mapping of period are points of the dynamical plane such that where is the smallest positive integer for which the equation holds at that z. We can introduce a new function: so periodic points are zeros of function : points z satisfying which is a polynomial of degree The degree of the polynomial describing periodic points is so it has exactly complex roots (= periodic points), counted with multiplicity. The multiplier (or eigenvalue, derivative) of a rational map iterated times at cyclic point is defined as: where is the first derivative of with respect to at . Because the multiplier is the same at all periodic points on a given orbit, it is called a multiplier of the periodic orbit. The multiplier is: a complex number; invariant under conjugation of any rational map at its fixed point; used to check stability of periodic (also fixed) points with stability index A periodic point is attracting when super-attracting when attracting but not super-attracting when indifferent when rationally indifferent or parabolic if is a root of unity; irrationally indifferent if but multiplier is not a root of unity; repelling when Periodic points that are attracting are always in the Fatou set; that are repelling are in the Julia set; that are indifferent fixed points may be in one or the other.